When the capacitor is connected to the DC voltage source, initially the positive terminal of the DC supply pulls the electrons from one terminal and pushes the electrons to the second terminal. This process continues until one plate is positively charged and the other plate is negatively charged. At this point, the plates of the capacitors are saturated and there will not be any current flow. The capacitor now behaves as an open circuit. Now if the DC voltage is increased the capacitor will get damaged.
When an AC voltage is connected to the capacitor initially the plates get charged. Later when the direction of the AC supply changes the capacitor will discharge. This process continues due to the change in supply voltage